QUANTIZATION MATRIX FOR MEDICAL IMAGE COMPRESSION USING FRAMELET TRANSFORM

In this paper, two quantization matrices are proposed that is suitable to compress medical images using framelet transform. Also two algorithms are suggested to compress medical images. One of them is used for grayscale and color medical images while the second is used for grayscale medical images only. It is found that the first proposed quantization matrix is better than the second in terms of Peak Signal to Noise Ratio (PSNR). While the second proposed quantization matrix is faster than the first. The work suggested in this paper is compared with wavelet and multiwavelet based algorithms and other previously related works and it is found that the quantization matrices proposed are most suitable for compression medical images with framelet transform and framelet transform is the best compression method for medical images.


Introduction
Following the rapid development of information and Communication Technologies, more and more information has to be processed, stored, and transmitted in high speed over networks.The need for data compression and transmission is increasingly becoming a significant topic in all areas of computing and communications.Computing techniques that would considerably reduce the image size that occupies less space and bandwidth for transmission over networks form an active research [Fatima B. Ibrahim,2010].
Image compression plays a critical role in telematics applications and especially in telemedicine as shown in Figure 1.Instance, it is necessary that medical images be transmitted so as that reliable, improved and fast medical diagnosis performed by many centers could be facilitated.To this end, image compression is an important research issue.The difficulty, however, in several applications lies on the fact that, while high compression rates are desired, the applicability of the reconstructed images depends on whether some significant characteristics of the original images are preserved after the compression process has been finished [Adina Arthur and V .Saravanan,2012].

Fig. 1 Telemedicine Concept [Adina Arthur and V .Saravanan,2012].
For instance, in medical image compression applications, diagnosis is effective only when compression techniques preserve all the relevant and important image information needed.This is the case with lossless compression techniques.Lossy compression techniques, on the other hand, are more efficient in terms of storage and transmission needs but there is no warranty that they can preserve the characteristics needed in medical image processing and diagnosis.In this latter case, of lossy compression, image characteristics are usually preserved in the coefficients of the domain space in which the original image is transformed.That is, for instance, in the Discrete Wavelet Transform (DWT) based medical image compression, the wavelet coefficients keep all the information needed for reconstructing the medical image [J.Pinto and Jayanand P. Gawande,2012].
Framelet is very similar to wavelets but has some important differences.Framelet has two or more high frequency filter banks, which produces more sub bands in decomposition.This can achieve better time-frequency localization ability in signal processing.Moreover, framelet is more robust [Runhai Jiao and Biying Lin,2010].In this paper, a new medical image compression algorithm is proposed using framelet transform.

Framelet Transform (FLT)
FLT is based on the theory of multi-resolution analysis (MRA) [RitamMisra .etal,2012] and is an extension of wavelet transform in the sense that it is defined in terms of one scaling function given by eq. ( 1): and two wavelet functions given by eq. ( 2): Where , i = 0, 1, 2, are the filters that follow Perfect Reconstruction (PR) conditions.This implies that the synthesis filters are time flipped versions of the analysis filters.

Quantization Matrix
Quantization is the process of reducing the number of bits needed to represent the transformed coefficients by reducing the precision of those values by dividing each element in the transformed image matrix D by the corresponding element in the quantization matrix, and then rounding to the nearest integer value as illustrated in eq.(3).
Since this process is a many-to-one mapping, it is a lossy process and it is the main source of compression in the encoding path [S.Taubman, et al,2002].A quantizer can be specified by its input partitions and output levels (also called reproduction points).If the input range is divided into levels of equal spacing, then the quantizer is termed as a uniform quantizer, and if not, it is termed as a non-uniform quantizer iii.Quantization matrix in eq. ( 6) [M.Siddeq,2010]: Where the parameter is computed by selecting maximum coefficient from a frequency components and then multiply by ratio=75%.

Proposed Work
The suggested work in this paper can be divided as: i. Proposed Quantization Matrices a.The first proposed quantization matrix ( ) in eq. ( 7): The parameters and ( ) are computed by selecting maximum coefficient from a frequency components (Max ( ) and then multiply by ratio.Where = Max ( , = Max ( , and these ratios ( and ) are adjusts from the user to change the image quality and compression ratio.
b.The second proposed quantization matrix ( ) in eq. ( 8): Where, the parameter computed by selecting maximum coefficient from a frequency components multiplied by ratio (r) such as 0.05, 0.1, 0.5 etc., etc. and etc… These values are adjusts by the user to change an image quality to obtain better compression ratio.

ii. Proposed Algorithms a. The First Proposed Compression Algorithm
The block diagram of the first proposed compression algorithm shows in Figure 3.
1. Generate LL sub-band by applying FLT on image.

2.
Apply Quantization matrix on each block of the low-frequency sub-band (LL).

3.
Before compress an image using coding methods, the image quality must be saved, by Applying inverse quantization matrix for the sub-band which is resulted from step (2).

4.
The image quality is obtained from the difference between the original image and the reconstructed image.

5.
Compress the ( ) after quantization process, using RLE and Huffman coding.

6.
Compress the D-Matrix resulting from step (4), by using Arithmetic Coding.

7.
To compress D-Matrix, the matrix should be dividing by a value called Q-Value.This value is adjusted by the user to change the image quality to obtain compression ratio.The range of the "Q-Value" between {1…m}, where m represents maximum value in D-Matrix as shown in eq. ( 9): Where is ratio such as (0.01, 0.02,…0.05,0.1, 0.2,…0.5,etc.).Then minimized D-Matrix using (Minimize Algorithm [M.Siddeq,2010]) and then compress each row by arithmetic coding.

 Decompression System of Algorithm (1)
The decompression algorithm will be the inverse for the compression algorithm, and the (Sequential Search Algorithm [M.Siddeq,2010]) must be used to construct D-Matrix then add with LL reconstructed to increase the quality as shown in Figure 4.

Fig. 4 Decompression Algorithm (1). b. The Second Proposed Compression Algorithm
The main reason of using Framelet transform is to reduce an image dimensions, and the high-frequencies coefficients are ignored (i.e.not used in algorithm ( 1)), this process increases compression ratio.But this will affect on the quality of some images, especially images that do not contain high psychovisual redundancy.Therefore, this algorithm suggests compressing each component to save quality for the medical images.
The block diagram of the proposed algorithm ( 2) is shown in Figure 5.

Fig. 5 Encoder of Compression Algorithm (2).
The procedure of algorithm (2) can be explained in the following steps: 1.
The pixels of an image are organized in groups of pixels and each group is compressed separately.If the number of image rows or columns is not a multiple of 16, the bottom row and the rightmost column are padded with zeros as many times as necessary.

2.
A single-stage Framelet transform is then applied on each group of pixels to create an map of nine frequency bands 3.
Each of the 576 frequency components in a map are divided by a separate numbers called Quantization matrix, this quantization matrix represented by the equations ( 7) or (8) and then rounded to an integer.

4.
Finally, the same procedure of algorithm (1) should be used to obtain the difference matrix between original image and reconstructed image, as shown in Figure 5 then compress the D-Matrix for adding it's consequentially in decompressed process to save image quality.

 Decompression System of Algorithm (2)
The decompression algorithm represents the inverse for each process in algorithm (2), and the Sequential Search Algorithm [M.Siddeq,2010]must be used to construct D-Matrix, then add with the reconstructed image to increase the quality as shown in Figure 6.

Fig. 6 System Model of Decompression Stage of Algorithm (2). c. Compress and Decompress Color Images
This algorithm is proposed for compressing the color medical images.First the colors images are converted into form, then applying algorithm (1) on each layer independently, this means each layer from are compressed as a grayscale image.Figure 7 show that algorithm (1) is applied on each layer.

Fig. 7 Layers are Converted to Layer, and then Compressed by Algorithm (1).
For decompression color images, apply decompression on each layer then collect all layers in one matrix and convert format to color image.Describing the comparison between the original image and the decompressed image using algorithm (1) is shown in Figure 9: Finally testing the proposed algorithm (1) on a number of colour medical images of different types.The comparison between original and decompressed image is shown in Figure11.

Fig. 11 Comparison between Original Colour Medical Test Image and its Decompressed Image, (a) Original US (832×832) pixel Size = 2028 Kbytes, (b) Decompressed US image
Compressed image Size= 127.304Kbytes.
Good image quality and good compression ratio for the color medical images are illustrate in Table 4 after applying algorithm (1).

Fig. 10
Fig. 10 Comparision between Original Image and Decompressed Image (a) Original US.bmp Size= 49.3623 Kbytes, (b) Decompressed US Compressed image Size = 9.6599 Kbytes

Table 2
illustrates that the PSNR and CR for different images using FLT is better than DWT and DMWT.